Random projection algorithms for convex set intersection problems

The focus of this paper is on the set intersection problem for closed convex sets admitting projection operation in a closed form. The objective is to investigate algorithms that would converge (in some sense) if and only if the problem has a solution. To do so, we view the set intersection problem as a stochastic optimization problem of minimizing the "average" residual error of the set collection. We consider a stochastic gradient method as a main tool for investigating the properties of the stochastic optimization problem. We show that the stochastic optimization problem has a solution if and only if the stochastic gradient method is convergent almost surely. We then consider a special case of the method, namely the random projection method, and we analyze its convergence. We show that a solution of the intersection problem exists if and only if the random projection method exhibits certain convergence behavior almost surely. In addition, we provide convergence rate result...

• Type: Journal
• Year: 2010
• Where: CDC
• Authors: Angelia Nedic
• Keywords: CDC 2010, Control Systems, Intersection Problem, Set Intersection Problem, Stochastic Optimization Problem